In particular, an horizontal area of the screen (see picture) does not react anymore to my finger, causing the tablet to be hardly usable in tablet mode!

Screen area where the finger touch is no more felt

Possible reasons for touchscreen problems can be (as in other contexts) software or hardware.
In the first case it could be sufficient to try one of the possible troubleshooting solution listed in the following post available in the Microsoft Community forum: Surface touchscreen: Problems with touch. The first solution is very simple, and it consists to properly clean the screen, … but some of the following solutions are more intrusive and possibly require a reset of the device in order to solve the software problem!

Therefore before starting with more intrusive possible solutions, it can be convenient to understand if it is really a software problem or, on the contrary, it is an hardware one that necessarily requires so send the device to a repair centre (hoping that it is still under warranty, that is 2 years long).
A further trial can be to download the Sony calibration tool (note: in fact, Surface Pro screens are made by Sony) and use it to fix the problem, as suggested from this post.

If the problem persists, to easily check if it is a HW problem, you can boot the Surface 3 Pro in BIOS mode (power + volume plus buttons pressed together for about 20 seconds) and then check if, even without Windows S.O. running, the same screen zone continues to not respond to a finger touch (so not opening the windows for choosing the related BIOS setup options): in that case please check if your Surface is still under warranty and (especially if the answer is yes) contact by phone the Customer service! My personal experience with that support was great and in a few minutes I received by email the label to sent (free of charge) my device to the repair centre in Germany.

On the contrary, if, with the BIOS screen the touchscreen properly works, it is surely a SW problem … so go on to try the possible solutions suggested in the previously described post!

Boot in BIOS mode (power and volume plus buttons pressed for 20 seconds) and verify that the touch problem persists also without Windows running

For asking customer support you can use the following links:

Device repair : from here you can register your device inserting the serial number and then, clicking on it (and waiting for a while), see when the manufacturer warranty ends.